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Optimal Trajectory Planning for a Single Train

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Optimal Trajectory Planning and Train Scheduling for Urban Rail Transit Systems

Part of the book series: Advances in Industrial Control ((AIC))

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Abstract

In this chapter, the optimal trajectory planning problem for the operation of a single train under various constraints and with a fixed arrival time is considered. The objective function corresponds to a trade-off between the energy consumption and the riding comfort . Two approaches are proposed to solve this optimal control problem, viz. a pseudospectral method and a mixed integer linear programming (MILP) approach. In the pseudospectral method, the optimal trajectory planning problem is recast into a multiple-phase optimal control problem, which is then transformed into a nonlinear programming problem. For the MILP approach, the optimal trajectory planning problem is reformulated as an MILP problem by approximating the nonlinear terms by piecewise affine functions. The performance of these two approaches is compared through a case study. The work discussed in this chapter is based on Wang et al. (Proceedings of the 14th international IEEE conference on intelligent transportation systems (ITSC 2011). Washington DC, USA, pp 1598–1604, 2011) [1]; Wang et al. (Transp Res Part C 29:97–114, 2013) [2]; Wang et al. (Proceedings of the 13th IFAC symposium on control in transportation systems (CTS’2012). Sofia, Bulgaria, pp 158–163, 2012) [3].

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Notes

  1. 1.

    A limiting gradient is defined as the maximum railway gradient that can be climbed without the help of a second power unit.

  2. 2.

    The transformation from \(\frac{\mathrm {d}v}{\mathrm {d}t}\) to \(\frac{\mathrm {d}\tilde{E}}{\mathrm {d}s}\) goes as follows:

    $$\begin{aligned} \frac{\mathrm {d}v}{\mathrm {d}t}=\frac{\mathrm {d}v}{\mathrm {d}s}\frac{\mathrm {d}s}{\mathrm {d}t}=v\frac{\mathrm {d}v}{\mathrm {d}s}=\frac{\mathrm {d}\tilde{E}}{\mathrm {d}s}, \end{aligned}$$

    where \(\tilde{E}=0.5v^2\).

  3. 3.

    The transformation from \(u v \mathrm {d}t\) to \(u \mathrm {d}s\) goes as follows:

    $$\begin{aligned} u \cdot v \text { } \mathrm {d}t=u \frac{\mathrm {d}s}{\mathrm {d}t} \mathrm {d}t = u \text { } \mathrm {d}s. \end{aligned}$$

    In addition, the transformation from \(\Big \vert \frac{\mathrm {d}u}{\mathrm {d}t} \Big \vert \mathrm {d}t\) to \(\Big \vert \frac{\mathrm {d}u}{\mathrm {d}s} \Big \vert \mathrm {d}s\) goes as follows:

    $$\begin{aligned} \Big \vert \frac{\mathrm {d}u}{\mathrm {d}t} \Big \vert \mathrm {d}t = \Big \vert \frac{\mathrm {d}u}{\mathrm {d}s} \Big \vert \Big \vert \frac{\mathrm {d}s}{\mathrm {d}t} \Big \vert \mathrm {d}t = \Big \vert \frac{\mathrm {d}u}{\mathrm {d}s} \Big \vert \mathrm {d}s, \text { if } \frac{\mathrm {d}s}{\mathrm {d}t} >0. \end{aligned}$$
  4. 4.

    The approximation error can be reduced by taking more regions.

  5. 5.

    For M affine subfunctions with \(M>3\) a similar procedure can be used.

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Authors and Affiliations

  1. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China

    Yihui Wang & Bin Ning

  2. Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands

    Ton van den Boom & Bart De Schutter

Authors
  1. Yihui Wang
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  2. Bin Ning
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  3. Ton van den Boom
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  4. Bart De Schutter
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Correspondence to Yihui Wang .

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Wang, Y., Ning, B., van den Boom, T., De Schutter, B. (2016). Optimal Trajectory Planning for a Single Train. In: Optimal Trajectory Planning and Train Scheduling for Urban Rail Transit Systems. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-30889-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-30889-0_3

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